Quantitative Study of Plasmonic Gold Nanostar Geometry Toward Optimal SERS Detection

Complex shapes of gold nanostars (GNS) have been the major obstacle to the comprehensive understanding of the rich plasmonic phenomena of GNS. We conducted extensive, systematic numerical study of far-field and near-field spectral responses of GNS for varying geometrical parameters (e.g., number of spikes, spike tip angle, and spike-to-core ratio) to elucidate the relationship between the optical property and the geometry of GNS. We found that symmetric configurations of GNS spikes generated both dipole and quadrupole localized surface plasmon (LSP) modes, their hybridization resulted in the final near-field intensity at the spike tips, and there existed a specific GNS geometry that optimized the hybridization and thus the E-field intensity for any given wavelength. From these results, we further identified the shapes of GNS that generated the strongest Surface Enhanced Raman Scattering (SERS) enhancement factors in the visible and NIR spectral range. Hence, our results provide guiding principles in determining the optimal geometry of GNS for SERS applications with any choice of excitation wavelength.


Introduction
Gold nanostars have emerged as a key nanotransducer for a variety of cutting-edge nanophotonic applications, such as label-free chemo/biosensing via surface-enhanced Raman spectroscopy (SERS), ultrahigh-resolution imaging, cancer diagnostics, and photothermal treatment [1][2][3][4][5]. In particular, GNS is of much interest as a promising plasmonic nanostructure due to its distinctive characteristic of huge electromagnetic (EM) enhancement near the edge of subnanoscale spikes, resulting from hybridization of plasmons between a core and spikes [6]. Besides, owing to the large geometric degrees of freedom, GNS features wider range of spectral responses in both visible and near-infrared (NIR) [7] wavelength than conventional gold nanospheres. The optical responses of GNS are primarily attributed to the dispersive nature of noble gold metal and also the surface plasmon polaritons (i.e., evanescently decaying surface waves) [8]. These factors are intertwined with the complex shape of GNS to result in unique polarizability that sensitively depends on the wavelength and GNS geometry [2,[9][10][11].
To make effective use of GNS in diverse applications, top-down nanofabrication and chemical synthesis method, mainly a seed-mediated growth, have been developed to control the geometry of GNS (e.g., spike length, number of spikes, and GNS size) [12][13][14][15]. Previous theoretical and numerical works demonstrated sensitive dependence of plasmonic behaviors of GNS on the particle geometry [6,11,16], which was directly associated with SERS enhancement factors (EF). The SERS EF of GNS have been reported in the range of 10 7 on the paper substrate [14]. Besides, even higher SERS EF (~ 10 9 ) have been experimentally demonstrated using arrays of GNS [17,18]. Compared to the moderate SERS EF (~ 10 5 ) of conventional gold nanoparticles [19,20], these high SERS EF of GNS open a new door to the molecular detection of ultralow concentration or even a single molecule. However, despite the great interest in GNS for their practical applications, systematic and comprehensive studies to unravel the interdependence between optical properties and complex geometry of GNS are still lacking. Accordingly, one of the most basic but practically critical questions such as follows has not been clearly addressed yet: what particular GNS geometry is the most suitable for generating the strongest local EM field through optimal hybridization of plasmons for a given wavelength? To provide a deeper insight into this problem, we conducted extensive numerical study on the optical properties of GNS over a diverse range of geometric parameters of GNS. Our work elucidates how the optimal hybridization of multiple localized surface plasmon (LSP) modes leads to the strong enhancement of local E-field in GNS and how the geometry of GNS affects such hybridization. Our results show that the SERS EF of GNS, when the shape is optimally designed, can even reach up to 10 11 theoretically, which is strong enough for reliable SERS-based molecular detection in the visible and NIR range.

Methods
In this work, both far-field and near-field spectral responses of GNS-under the illumination of linearly polarized monochromatic plane waves-are calculated using a threedimensional (3D) finite-difference time-domain (FDTD) modeling method (Lumerical FDTD solutions, S2019A-R1). A chemically synthesized GNS is typically shaped as a 3D object with cone-shaped spikes radially protruding out of a spherical core in every or arbitrary direction. However, as in the case for nanorods/nanowires [21,22] and also for top-down nanofabricated GNS [12], the localized surface plasmon resonance (LSPR) of highly anisotropic gold nanoparticles gets immensely maximized when the extended dimension is in parallel with the electric field of the incoming EM wave. This implies that the subset of GNS spikes that lie on the plane perpendicular to the propagation direction of the incoming wave will play the dominant role in determining the optical response of GNS at LSPR [6,9,11,16,23]. Therefore, we focus on the planar GNS geometry (supplementary Fig. 1) to capture the essence of plasmon hybridization while avoiding otherwise overly complicated geometric degrees of freedom of GNS. Moreover, given that planar GNS can be actually fabricated in various shapes with nanofabrication methods [11,12,16,17], its systematic numerical study is of direct relevance to practical applications.

Far-Field Spectral Response
First, we have investigated the far-field characteristics of GNS using extinction spectra (see the supplementary method for the details). Figure 1a-h show how the far-field spectral response of Rayleigh particles (e.g., particle size R out = 40 nm, thickness D = 40 nm) depends on SCR and N spike , the two primary parameters that govern the level of anisotropy, i.e., the spike length from the core surface and the tip angle (top panel of Fig. 1, and supplementary Fig. 5). For a large value of SCR (e.g., SCR = 4), which corresponds to spike-dominant GNS, the extinction spectra show a single major peak which moves to longer wavelength from ~ 675 to ~ 830 nm as N spike increases (Fig. 1a). To investigate how much this feature can be ascribed to the two collinear spikes aligned along the E-field, we have calculated extinction spectra for nanodiamonds made of bottom-to-bottom-faced two nanospikes (or equivalently nanotriangles in our case) with the identical tip angle as for the corresponding GNS of N spike = 4 and 16, respectively (Fig. 1b, c). Nanodiamonds capture the overall shape of GNS extinction spectra fairly well. The trend of spectral red-shift in the increase of N spike is also well explained by the elevated anisotropy of nanodiamonds as the spike tip angle gets smaller accordingly. However, the nanodiamond spectra are more red-shifted than GNS, for the large values of N spike (Fig. 1c). Interestingly, the nanodiamond spectrum is blue-shifted to some degree if a cylindrical core is introduced inside the nanodiamond (Fig. 1d). The presence of bulky cylindrical core as well as extra spikes therefore explains the blue-shift of GNS spectra from the corresponding nanodiamonds. For a smaller value of SCR (e.g., SCR = 2.5), which corresponds to core-dominant GNS, the extinction spectra show double major peaks at ~ 600 nm and ~ 660 nm for N spike = 4 in contrast to a single major peak of spike-dominant GNS (Fig. 1e). Likewise, the corresponding nanodiamond shows a very similar spectrum (Fig. 1f). The spike tip angle (~ 44°) in this case implies that the nanodiamond is made of two almost equilateral triangles. The appearance of spectral double peaks by two bottom-to-bottom-faced nanotriangles was previously demonstrated [24] due to longitudinal and transverse hybridizations between two triangles. The increase of N spike makes the nanodiamond slimmer, which suppresses the transverse hybridization (i.e., the shorter wavelength peak) while strengthening the longitudinal mode of hybridization (Fig. 1e). The spectral blue-shift from nanodiamond by the core and extra spikes of GNS is also apparent (Fig. 1g, h). As compared to spike-dominant GNS (e.g., SCR = 4), the blue-shifting effect of extra spikes appears more substantial for core-dominant GNS (Fig. 1d, h). Interestingly, the larger core size in core-dominant GNS (e.g., SCR = 2.5) additionally allows higher order surface plasmon modes for large N spike , yielding broad extinction spectra and multiple spectral peak shoulders (Fig. 1g, h). The spectral redshift in N spike was experimentally observed for chemically synthesized surfactant-free GNS [25]. This facile spectral tunability renders GNS useful for applications in a broad visible and NIR range.
We then have examined the dependence of spectral response on particle size. As the size of four-spike GNS increases from Rayleigh regime to quasi-Mie regime, the extinction spectra of core-dominant GNS (SCR = 2.5) show extinction enhancement and spectral broadening/red-shift exclusively for the longitudinal hybridization mode (i.e., Fig. 1 Far-field extinction spectra of spike-dominant GNS (SCR = 4; a-d) and core-dominant GNS (SCR = 2.5; e-h) in Rayleigh regime (particle outer radius R out = 40 nm and thickness D = 40 nm). a and e Dependence on N spike . b and f Comparison between N spike = 4 GNS and the analogous nanodiamond with same tip angle and particle size. c and g Comparison between N spike = 16 GNS and the analogous nanodiamond. d and h Comparison between N spike = 16 GNS and the analogous nanodiamond with cylindrical core. Spectral dependence on particle size with SCR fixed to 2.5 for N spike = 4 GNS and the analogous nanodiamond (i and j) and N spike = 16 GNS and the analogous nanodiamond (k and l) longer wavelength peak) ( Fig. 1i; supplementary Fig. 6). These features, with mostly being reproduced by the corresponding gold nanodiamonds (Fig. 1j), are ascribed to the reliance of light scattering and absorption on the size and the shape of GNS by way of the physics of damped harmonic oscillator model [26]. For N spike = 16, the additional spikes as well as the core of GNS further manifest more complex resonance behavior [27] in the extinction spectra, as compared to nanodiamonds (Fig. 1k, l).

Near-Field Spectral Response
Next, we have investigated the near-field responses of GNS that is directly associated with SERS performance. Unlike far-field response that mainly depends on dipole electric field, near-field optical response is governed by multipole field properties [26] as well. Electric field is greatly enhanced at the tip of GNS spikes but its exact strength and distribution for any given GNS geometry appear quite sensitive to the wavelength (supplementary Figs. 7 and 8 for N spike = 4 and N spike = 8). To better understand how the near-field enhancement depends on wavelength, we have calculated the local maximum field intensity in the vicinity of GNS spike edges, where plasmonic hotspots are located, for spike-dominant GNS in Rayleigh regime (SCR = 4, R out = 40 nm, and D = 40 nm). In case of N spike = 4, the field intensity at the tips of two longitudinal spikes is dominant over vertical spikes and shows a primary peak at ~ 830 nm and a secondary peak at ~ 730 nm (Fig. 2a). To examine the E-field component that is enhanced exclusively by the localized surface plasmon, we have calculated in-plane distribution of axial component of electric field Re(E z ), i.e., the transverse magnetic (TM) mode [8]. This has revealed quadrupole-mode structure at the 830-nm peak, in contrast to dipole mode at the 730-nm peak (Fig. 2a). The shorter wavelength dipole mode is responsible for the far-field extinction spectra as in Fig. 1, whereas the longer wavelength quadrupole mode dominates the near-field enhancement. The quadrupole mode is generated by an image dipole [28] and this dual-mode structure of N spike = 4 GNS originates from the analogous nanodiamond although the nanodiamond spectrum is red-shifted (Fig. 2a). In case of N spike = 8, we have studied the near-field spectrum at (two) longitudinal spikes and (four) diagonal spikes separately (Fig. 2b, c). The hotspots at longitudinal spikes feature the dual-mode structure (dipole mode at ~ 650 nm and quadrupole mode at ~750 nm; Fig. 2b) similarly as Fig. 2a. Moreover, N spike = 4 GNS with the same tip angle as N spike = 8 GNS reproduces the longitudinal tip spectrum of N spike = 8 almost identically. On the other hand, the hotspots at the diagonal spikes also show two spectral peaks at ~650 nm and ~700 nm and the plot of Re(E z ) further reveals dipole and quadrupole modes at these wavelengths, respectively (Fig. 2c). Besides, 45°-oriented N spike = 4 GNS with the same tip angle as N spike = 8 GNS well reproduces the diagonal tip spectrum of N spike = 8 GNS. These results altogether indicate that the near-field optical response of complex GNS geometry such as 8-spike GNS on the whole can be ascribed to the dipole and quadrupole modes that arise from more fundamental symmetry groups (i.e., 4-spike GNS that are aligned either in parallel or at 45° with respect to the incoming wave polarization). Figure 2a, b also show blue-shift of the primary peaks as N spike increases from 4 to 8. To investigate whether this feature can be largely ascribed to the decrease in spike tip angle (supplementary Fig. 5) that accompanies the increase of N spike for a fixed SCR value, we have calculated near-field spectra of nanodiamonds for varying spike tip angle and wavelength (supplementary Fig. 9). The quadrupole-mode wavelength is indeed blue-shifted as the nanodiamond spike tip angle decreases while the dipole mode wavelength is red-shifted, which subsequently leads to more overlapping between the two modes. Notably, the two spectral peaks are even reversed in the positional order and intensity at small tip angles like 6° (i.e., dipole mode becomes the primary peak at longer wavelength; supplementary Fig. 9). Therefore, the major peak of near-field spectrum, generally speaking, represents the optimal hybridization of dipole and quadrupole modes, maximizing near-field intensity at a specific wavelength that is conventionally termed "LSPR wavelength" ( LSPR ). In contrast with monotonic red-shift of far-field extinction spectral peak, LSPR wavelength shows nonlinear dependence on N spike (supplementary Fig. 10a-d), with the blue-shift between N spike = 4 and 8 reflecting the feature of quadrupole mode in particular. Interestingly, LSPR wavelength shows biphasic increase in SCR for a fixed value of N spike (supplementary Fig. 10e), and this highlights the important role of GNS core size as well as spike tip angle in the optimal hybridization of different LSP modes.

SERS Enhancement Factor
Strong electric fields at the tips of GNS spikes result in substantial enhancement of Raman signal. To quantify the enhancement, we have calculated "SERS EF" by averaging the fourth power of local electric field over the surface of a nanostructure, i.e., where E 0 is the incident electric field magnitude and E is the enhanced electric field magnitude on the nanostructure surface [9,29,30]. Although SERS EF gets large near LSPR wavelength, the wavelength at which SERS EF is maximized is different than LSPR wavelength, rigorously speaking (supplementary Fig. 11). Therefore, we introduce "SERS wavelength" ( SERS ) to indicate the wavelength that maximizes SERS EF for a given GNS geometry. We have calculated optimal SERS EF at the corresponding SERS for each GNS structure, and then compared the values among Re(E z ) reveals dipole and quadrupole modes. The nanodiamond with the same tip angle and size shows similar behavior at its own LSP wavelengths (1′, 2′). b Spectra at the longitudinal spike tips of 8-spike GNS, 4-spike GNS, and nanodiamond of the same tip angle and size. c Spectra at the diagonal spike tips of 8-spike GNS and 4-spike GNS of the same tip angle and size different geometry of GNS. As is evident from Fig. 3, SERS EF of GNS is peaked at a specific value of SCR for a given particle size and N spike . For instance, SERS EF of 40-nm GNS with four spikes (N spike = 4) reaches the maximum value (~ 2 × 10 11 ) at SCR = 2.5 (Fig. 3a). Very similar curves of SERS EF are observed for the simplified gold nanostructures, i.e., nanodiamonds with or without cylindrical core, as in the case of far-field extinction spectra. All these results imply that there exists a golden ratio between the contributions from core and spikes for optimal plasmon hybridization.
As SCR gets larger than the optimal value, the more extended spike length as well as the more sharpened tip angle might be predicted to further enhance electric field from the aspect of depolarized lightning rod effect [31]. However, too slim and sharp spike cannot afford enough surface charge density required for the activation of localized surface plasmon polaritons [32]. Besides, the significantly shrunken volume of the core for large SCR further faces difficulty in supplying enough oscillating electrons for hybridization of core and spikes [33]. The increased anisotropy of the spike also shifts the LSPR wavelength more toward NIR (supplementary Fig. 10e), which exacerbates mechanical damping of conduction electron inside gold metal (i.e., the imaginary part of complex dielectric permittivity of gold increases for longer wavelength in red and NIR range) [34] (supplementary Fig. 12). These all together contribute to the decrease of SERS EF for large SCR values. As SCR gets smaller than the optimal value, on the other hand, more obtuse, shorter spikes make GNSs appear as smoother and more isotropic objects, which abolish the role of spikes as field-enhancing lightning rods. Thus, SERS EF decreases in this case too (supplementary Fig. 13).
Another discernible feature in Fig. 3b is the decrease of SERS EF as N spike increases, with the degree of reduction being moderate for N spike = 8 but substantial for N spike = 12 and 16. This can be ascribed to the spectral broadening due to high-order plasmon hybridization [28] as well as the increased dissipative damping from longer perimeter of GNS with more spikes [35]. It is also noticeable that 8-spike, 40-nm GNS reaches the maximum SERS EF at the same optimal SCR value (~ 2.5) as 4-spike GNS (Fig. 3b). Given that the spike tip angle (~ 31°) of 8-spike GNS is smaller than the angle (~43°) of 4-spike GNS, it is presumed that the enhanced lightning rod effect counterbalances the effect of spectral broadening and dissipative damping as N spike increases from 4 to 8, while maintaining the optimal SCR at the similar values. These characteristics of SERS EF dependence on SCR and N spike overall persist even as the size of GNS gets bigger (Fig. 3c-e). Compared to 40-nm GNS, however, larger particle size (60 nm and 80 nm) moves the optimal SCR to higher values (2.75 and 3) for N spike = 4, which we ascribe to the larger core volume that can uphold enough number of oscillating electrons to support hybridized plasmon modes even for sharper and narrower spikes. Besides, the larger particle size and higher SCR value of GNS prompt extinction peak as well as SERS wavelength toward NIR (supplementary Fig. 10), extending the workable wavelength range for SERS application using GNS (Fig. 3c, d). Remarkably, as the particle size further increases to 100 nm, SERS EF for N spike = 4 is surpassed by N spike = 8, reaching the maximum near SCR = 2.5 (Fig. 3e). The E-field distribution reveals that the core of 100-nm GNS is big enough to sustain/provide oscillating electrons even to the subsidiary spikes that are not parallel to E-field, thus extending the degree of lightning rod effect [36]. However, the SERS EF of 100-nm GNS is overall decreased as compared to smaller GNS size (e.g., 40 nm, 60 nm, and 80 nm), which is due to diminishing LSPR effect for particles of large size comparable to the wavelength of incident light [8]. Fig. 3 Dependence of optimal SERS EF on SCR, N spike , and size of GNS. a Comparison between 4-spike GNS and 2-spike nanodiamond (with or without core) for R out = 40 nm. Middle and right panels show E-field intensity 2D distribution of the nanodiamonds at their own SERS wavelengths. b-e Dependence on N spike for R out = 40-, 60-, 80-, and 100-nm GNS. Middle and right panel show E-field intensity 2D/1D distributions at the wavelength that maximizes SERS EF for each particle size, i.e., the global peak point in the left panel. 2D distributions are overlaid by the corresponding GNS shape. Inset image shows magnified view of E-field distributions at the edge of a spike. Particle thickness is identical to R out in all cases. Scale bar = 30 nm ◂ Fig. 4 Optimal SERS EF versus SERS wavelength in the visible-NIR wavelength range for various GNS shapes, providing guidance on selection of the most efficient GNS geometry for a given laser excitation wavelength In Fig. 4, we have replotted the results in Fig. 3 indicating the optimal SERS wavelength for each GNS geometry in order to elucidate the best shape and size of GNS for any given wavelength. In the visible range (i.e., Regions A and B), smaller size GNSs, such as 40 nm, yield better SERS EF than the bigger ones, with larger spike number (N spike = 8) being superior to smaller number (N spike = 4) in shorter wavelength (Region A) and vice versa for longer wavelength (Region B). Toward the NIR range (Regions C and D), bigger GNSs (80 nm) with N spike = 8 and 4 present the best SERS enhancement in a similar fashion. Overall, GNS provides reasonable SERS EF in the order of at least 10 9 up to 10 11 in optical range. This degree of tolerance for shape (i.e., ranging the value of SCR from 2 to 4) further highlights the predominant characteristic of GNSs as a robust and sensitive SERS probe.
Extra optical variables such as refractive index of surrounding medium also affect SERS EF. As the medium refractive index varies within the typical range of aqueous buffer solution, far-field extinction spectrum red-shifts and SERS EF slightly increases (supplementary Fig. 14). The increment of refractive index induces the nanolens effect [29] that enhances the confinement of plasmonic hotspots emanating from GNS spikes.

Conclusions
Our quantitative numerical study of symmetric planar GNS provides comprehensive insights into the relationship between GNS geometry and its near-field optical properties. Symmetric configurations of GNS spikes generate quadrupole LSP modes in addition to dipole modes (Fig. 2). When spike tip angle is not so small, the quadrupole modes dominantly contribute to the enhancement of E-field at the spike tips. As the spike tip angle gets smaller, the effect of quadrupole mode diminishes while its resonance wavelength blue-shifts at the same time. In contrast, the contribution of red-shifting dipole mode becomes more significant for small tip angles. For a given particle size, the strongest near-field intensity is achieved when the two LSP modes are hybridized optimally. Not only the spike tip angle but also other geometric parameters such as core size and spikes number play important roles in the optimal hybridization of LSP modes (Figs. 3 and  4). We anticipate that our principal findings will be applicable to the case of GNS with spherical core and conical spikes. Nevertheless, it will be an interesting future work to conduct a similar numerical study on spherical GNS geometry. We believe that our work provides deliberate design rationales to untangle complex GNS geometry for the development of highly sensitive SERS probes toward potential biophotonic and plasmonic applications.